New insights into the properties of matter at high baryon density and low temperature arise from the Quarkyonic hypothesis. The general features of Quarkyonic matter are discussed as well as the application of its theory to the description of neutron stars. An exactly solvable model, with two phases and a hard equation of state at high densities is presented. This theory is a free theory of nucleons, composed of quarks, with the Fermi exclusion principle applied for both nucleon and quark densities. It is exactly solvable for a specific choice of probability to find a quark inside a baryon as a function of quark momentum. The solution shows explicitly the formation of a shell of baryons surrounding a filled Fermi sea of quarks in the high density phase. The low density phase is a free Fermi gas. The theory is explicitly dual between quarks and nucleons.